No. of Pages: 7 No. of Questions: 10 MIDSUMMER EXAMINATIONS 2001 Subject PHYSICS, PHYSICS WITH ASTROPHYSICS, PHYSICS WITH SPACE SCIENCE & TECHNOLOGY, PHYSICS WITH MEDICAL PHYSICS Title of Paper MODULE PA224 ELECTROMAGNETIC FIELDS MODULE PA254 ELECTROMAGNETICS AND OPTIONS A MODULE PA304 ELECTROMAGNETICS AND OPTIONS B Time Allowed Three hours Instructions to candidates Answer ALL questions in SECTION A. Allow one hour. Section A carries 40 marks. Answer THREE questions ONLY in SECTION B. The marking scheme for each question in SECTION B is indicated by the numbers in square brackets in the right-hand margin. A separate answer book MUST be used for Sections A & B. A LIST OF PHYSICAL CONSTANTS IS ATTACHED SECTION A A1. A thin spherical conducting shell of radius a is centred at the origin of coordinates and has an electrostatic potential given by V = V 0 for r a and V = V 0a r for r > a with zero reference at infinity. The medium surrounding the shell is vacuum. Find an expression for the energy stored in the electrostatic field associated with this potential distribution. Note: In spherical coordinates a volume element dv = r 2 sinθ dr dθ dφ.
2. A2. i) Write down Maxwell s equations in terms of the electric field E and the magnetic induction B which are appropriate to a vacuum. Derive the wave equation for E in the vacuum. [You may assume the following vector identity, valid for any vector field F: curl( curlf)= ( divf) 2 F. ] A3. i) The vectors describing the electromagnetic field are E, D, B, and H. In terms of these quantities write down general expressions for a) the energy densities in the electric and magnetic fields W E and W B b) the Poynting vector N. What is the physical interpretation of the Poynting vector? In integral form Poynting's theorem can be written as W t ( E + W B )dτ + N.dS V + j f.edτ = 0 V, S where j f is the conduction current density, and S is the surface of an arbitrary volume V. Explain the physical significance of each term in this equation. What is the overall physical interpretation of the equation? A4. a) An electromagnetic wave propagating in a vacuum is incident upon another medium. State the boundary conditions for the electromagnetic wave vectors E, D, B and H. In each case identify the relevant Maxwell equation which you would use to determine this boundary condition. b) State the specific assumptions made in determining the boundary conditions for H and D.
3. SECTION B B1. The field B inside a long solenoid which is completely filled with a rod of magnetic material is given by B = B 0 + B M where B 0 is the field in the absence of the rod and B M arises due to the magnetization M of the rod. Use the idea of the magnetization surface currents existing on the rod to show that it is possible to introduce a magnetic intensity H which may be expressed in terms of B and M. [9] Show that the curl of H depends only on the motion of free charges. [3] A long solenoid which has 15 turns per cm and carries a current 0 1 A is wound on an even longer iron core which has a relative permeability µ r of 1000. Calculate the magnitude of the magnetization surface current density. [6] What would the magnetization current density have been if the solenoid was wound on a copper core with relative permeabilty µ r given by µ r 1 = 10 5? [2] B2. i) A parallel-plate capacitor consists of metal plates of area A and separation d, filled with a dielectric of relative permittivity ε, such that the capacitance C is given by C = εε oa d. The capacitor is charged to voltage V. Show that the energy stored in the electric field in the space between the plates is 1 2 CV 2. (Edge effects may be ignored.) [6] A long solenoid consists of N turns of wire wound uniformly on a solid cylinder of material of relative permeability µ. The cylinder is of length l and cross-sectional area a, such that the self-inductance L of the solenoid is given by L = µµ on 2 a. l The solenoid carries a steady current I. Show that the energy stored in the magnetic field within the solenoid is 1 2 LI 2. (You may assume that the magnetic field is uniform within the solenoid and of negligible magnitude outside.) [7]
4. i The charged capacitor in section (i) above is connected across a resistor and discharged, such that after a long time the energy that was stored in the capacitor is converted to heat in the resistor. Give a careful physical explanation of how the energy is transferred from the capacitor to the resistor. Illustrate your answer with diagrams as appropriate. [7] B3. i) The electric field of a plane electromagnetic wave propagating in a vacuum is given by E(z,t) = E o e j(ωt kz ) x ö. In which direction is the wave propagating? What is the wave vector of the wave? In which direction is the electric field polarised? [3] Substitute this expression for E(z,t) into an appropriate Maxwell equation to show that the magnetic field of the wave is B(z,t ) = E o c e j(ωt kz) ö y, where c is the wave phase speed. (You may assume that for such a wave field the gradient operator may be replaced by -jk, and the partial time derivative by jω.) [6] t i Show that the time-averaged energy flux in the wave is N = 1 2 ε o µ o E o2 ˆ z. where ε o and µ o are the permittivity and permeability of free space respectively. You may assume that the time-averaged value of cos 2 ωt is 1 ( 2 ). [6] iv) A radio transmitter emits radiation isotropically with a total radiated power of 10 kw. Estimate the amplitude of the electric and magnetic fields of the wave at a distance of 100 km. [5] (ε o = 8.85 10 12 F m 1 ; µ o = 4π 10 7 H m 1 )
5. B4. i) A plane electromagnetic wave propagates in a medium of relative permittivity ε, relative permeability µ, and electrical conductivity σ. Write down Maxwell s equations for this medium in terms of the electric field E and the magnetic field B. (You may assume that the free charge density ρ f in the medium remains zero.) [4] The medium is described as a "good electrical conductor" at angular frequency ω. What is meant by this statement? Show that in this case the following inequality is satisfied σ >> εε o ω. [4] i Derive the wave equation governing the electric field of the wave in this medium. [4] Note: you may assume the following vector identity, valid for any vector field F: curl( curlf) = ( divf) 2 F. iv) When this approximation is satisfied, the electric field of the wave is E(z,t) = E o e k c z e j(ωt k c z) x ö where kc = µµ oσω 2 What is meant by the skin depth δ of the wave? What is the value of δ at angular frequency ω? What is the phase speed v c of the wave at this angular frequency? [3] v) Show that in this approximation the phase speed of the wave is much less than the phase speed of a wave propagating in a non-conducting medium of the same ε and µ, given by v =1 µµ o εε o. What is the implication for the wavelength of the wave, relative to that of a wave of the same angular frequency propagating in the non-conducting medium? [5].
6. B5. An electromagnetic wave travelling in a vacuum is normally incident upon a dielectric of relative permittivity ε. Using the two boundary conditions for tangential fields, derive expressions for the ratio of the amplitude of the reflected electric field to the amplitude of the incident electric field and the ratio of the amplitude of the transmitted electric field to the amplitude of the incident electric field. You may assume that the relative permeability of the dielectric µ is 1. [8] Using the time average over a cycle of the Poynting vectors for the incident and reflected waves, demonstrate that the reflection coefficient R and the transmission coefficient T are given by R = 1 n 2 1 + n and T = 4n, ( 1 + n) 2 where n is the refractive index of the dielectric. [6] A laser beam has a power of 100 W and a diameter of 1 mm. Find the peak values of the electric and magnetic fields of the laser beam in air. In an experiment the laser beam propagates through a glass block with a refractive index of 1.59. Find the peak values of the electric and magnetic fields of the laser beam in the glass. [6] B6. A plane wave in free space is incident on a dielectric medium of refractive index, n, at an angle θ to the normal. Waves are reflected back into free space and transmitted into the medium. The transmitted wave propagates at an angle φ to the normal. E is in the plane of propagation. Show that the ratio between the amplitude of the reflected wave E or, and the amplitude of the incident wave E oi, is E or E oi = ncosθ cosφ ncosθ + cosφ. [8] Explain with the aid of a diagram how a beam of unpolarised light may be split into separate beams of polarised light with the planes of polarisation orthogonal to each other. [8] Derive an expression for the Brewster angle, stating clearly what assumption is necessary about the sum of the angles θ and φ. [4]
7. PHYSICAL CONSTANTS elementary charge e = 1.602 10 19 C electron rest mass me = 9.109 10 31 kg proton rest mass mp = 1.673 10 27 kg neutron rest mass mn = 1.675 10 27 kg Planck constant h = 2π h = 6.626 10 34 J s speed of light in vacuum c = 2.998 10 8 m s 1 Boltzmann constant k = 1.381 10 23 J K 1 Bohr magneton µ B = 9.274 10 24 J T 1 Stefan constant σ = 5.671 10 8 W m 2 K 4 radiation constant a = 4σ / c = 7.564 10 16 J m 3 K 4 Avogadro number N A = 6.022 10 23 mol 1 gas constant R = 8.314 J mol 1 K 1 gravitational constant G = 6.673 10 11 N m 2 kg 2 permittivity of vacuum εo = 8.854 10 12 F m 1 permeability of vacuum µo = 4π 10 7 H m 1 1 atomic mass unit u = 1.661 10 27 kg 931.494 MeV 1eV = 1.602 10 19 J ASTRONOMICAL CONSTANTS mass of Sun M O = 1.989 10 30 kg radius of Sun R O = 6.960 10 5 km luminosity of Sun L O = 3.90 10 26 J s-1 one parsec pc = 3.086 10 16 m mean Earth-Sun distance 1 a.u. = 1.496 10 11 m mass of Earth M O + = 5.97 10 24 kg mean radius of Earth R O + = 6370 km